Which Statement Best Describes How To Determine Whether F ( X ) = 9 − 4 X 2 F(x)=9-4x^2 F ( X ) = 9 − 4 X 2 Is An Odd Function?A. Determine Whether 9 − 4 ( − X ) 2 9-4(-x)^2 9 − 4 ( − X ) 2 Is Equivalent To 9 − 4 X 2 9-4x^2 9 − 4 X 2 .B. Determine Whether 9 − 4 ( − X 2 9-4(-x^2 9 − 4 ( − X 2 ] Is Equivalent To

Introduction

In mathematics, a function is considered odd if it satisfies the condition $f(-x) = -f(x)$ for all $x$ in the domain of the function. This property is crucial in various mathematical operations, such as differentiation and integration. In this article, we will explore how to determine whether a given function is odd, using the function $f(x) = 9 - 4x^2$ as an example.

Understanding Odd Functions

An odd function is a function that satisfies the condition $f(-x) = -f(x)$ for all $x$ in the domain of the function. This means that if we replace $x$ with $-x$ in the function, the resulting expression should be equal to the negative of the original function.

Determining Whether a Function is Odd

To determine whether a function is odd, we need to check if it satisfies the condition $f(-x) = -f(x)$. Let's apply this condition to the function $f(x) = 9 - 4x^2$.

Step 1: Replace $x$ with $-x$

We start by replacing $x$ with $-x$ in the function $f(x) = 9 - 4x^2$. This gives us:

$f(-x) = 9 - 4(-x)^2$

Step 2: Simplify the Expression

Now, we simplify the expression $f(-x) = 9 - 4(-x)^2$. To do this, we need to expand the square term $(-x)^2$. Since the square of any real number is always non-negative, we can write:

$(-x)^2 = x^2$

Therefore, we can simplify the expression as follows:

$f(-x) = 9 - 4x^2$

Step 3: Compare with the Original Function

Now, we compare the simplified expression $f(-x) = 9 - 4x^2$ with the original function $f(x) = 9 - 4x^2$. We can see that they are equivalent.

Conclusion

Since $f(-x) = 9 - 4x^2$ is equivalent to $f(x) = 9 - 4x^2$, we can conclude that the function $f(x) = 9 - 4x^2$ is an even function, not an odd function.

Which Statement Best Describes How to Determine Whether $f(x) = 9 - 4x^2$ is an Odd Function?

Based on our analysis, we can conclude that the correct statement is:

A. Determine whether $9-4(-x)^2$ is equivalent to $9-4x^2$

This statement accurately describes the process of determining whether a function is odd. By replacing $x$ with $-x$ and simplifying the expression, we can check if the function satisfies the condition $f(-x) = -f(x)$.

Discussion

In this article, we have explored how to determine whether a function is odd using the function $f(x) = 9 - 4x^2$ as an example. We have shown that the function is actually an even function, not an odd function. By following the steps outlined in this article, you can determine whether any given function is odd or even.

Conclusion

In conclusion, determining whether a function is odd or even is a crucial step in various mathematical operations. By following the steps outlined in this article, you can easily determine whether a function is odd or even. Remember to replace $x$ with $-x$, simplify the expression, and compare it with the original function to check if it satisfies the condition $f(-x) = -f(x)$.

References

• [1] "Functions" by Khan Academy
• [2] "Odd and Even Functions" by Math Open Reference
• [3] "Determining Whether a Function is Odd or Even" by Wolfram MathWorld

Additional Resources

• [1] "Functions" by MIT OpenCourseWare
• [2] "Odd and Even Functions" by Purplemath
• [3] "Determining Whether a Function is Odd or Even" by Mathway
  Q&A: Determining Whether a Function is Odd or Even =====================================================

Introduction

In our previous article, we explored how to determine whether a function is odd or even using the function $f(x) = 9 - 4x^2$ as an example. In this article, we will answer some frequently asked questions related to determining whether a function is odd or even.

Q: What is the difference between an odd function and an even function?

A: An odd function is a function that satisfies the condition $f(-x) = -f(x)$ for all $x$ in the domain of the function. An even function, on the other hand, satisfies the condition $f(-x) = f(x)$ for all $x$ in the domain of the function.

Q: How do I determine whether a function is odd or even?

A: To determine whether a function is odd or even, you need to replace $x$ with $-x$ in the function and simplify the expression. If the resulting expression is equal to the negative of the original function, then the function is odd. If the resulting expression is equal to the original function, then the function is even.

Q: What is the significance of determining whether a function is odd or even?

A: Determining whether a function is odd or even is crucial in various mathematical operations, such as differentiation and integration. Odd and even functions have different properties and behaviors, and understanding these properties is essential in solving mathematical problems.

Q: Can a function be both odd and even?

A: No, a function cannot be both odd and even. If a function is odd, it satisfies the condition $f(-x) = -f(x)$, and if it is even, it satisfies the condition $f(-x) = f(x)$. These two conditions are mutually exclusive, and a function cannot satisfy both conditions simultaneously.

Q: How do I determine whether a function is odd or even using a graph?

A: To determine whether a function is odd or even using a graph, you can use the following method:

• If the graph of the function is symmetric with respect to the origin, then the function is odd.
• If the graph of the function is symmetric with respect to the y-axis, then the function is even.

Q: Can a function be odd or even if it is not defined at a point?

A: Yes, a function can be odd or even even if it is not defined at a point. The definition of an odd or even function only requires that the function satisfies the condition $f(-x) = -f(x)$ or $f(-x) = f(x)$ for all $x$ in the domain of the function, except possibly at a single point.

Q: How do I determine whether a function is odd or even using a calculator?

A: To determine whether a function is odd or even using a calculator, you can use the following method:

• Enter the function into the calculator and press the "graph" button.
• Use the "zoom" and "trace" features to examine the graph of the function.
• If the graph is symmetric with respect to the origin, then the function is odd.
• If the graph is symmetric with respect to the y-axis, then the function is even.

Conclusion

In conclusion, determining whether a function is odd or even is a crucial step in various mathematical operations. By following the steps outlined in this article, you can easily determine whether a function is odd or even. Remember to replace $x$ with $-x$, simplify the expression, and compare it with the original function to check if it satisfies the condition $f(-x) = -f(x)$ or $f(-x) = f(x)$.

References

• [1] "Functions" by Khan Academy
• [2] "Odd and Even Functions" by Math Open Reference
• [3] "Determining Whether a Function is Odd or Even" by Wolfram MathWorld

Additional Resources

• [1] "Functions" by MIT OpenCourseWare
• [2] "Odd and Even Functions" by Purplemath
• [3] "Determining Whether a Function is Odd or Even" by Mathway